UNIT AND UNITARY CAYLEY GRAPHS FOR THE RING OF EISENSTEIN INTEGERS MODULO \(n\)

Let \({E}ₙ\) be the ring of Eisenstein integers modulo \(n\). We denote by \(G({E}ₙ)\) and \(G_{{E}ₙ}\), the unit graph and the unitary Cayley graph of \({E}ₙ\), respectively. In this paper, we obtain the value of the diameter, the girth, the clique number and the chromatic number of these graphs. We also prove that for each \(n>1\), the graphs \(G(Eₙ)\) and \(G_{Eₙ}\) are Hamiltonian.